Abelian Category Research Articles

From n-exangulated categories to n-abelian categories

Herschend-Liu-Nakaoka introduced the notion of n-exangulated categories. It is not only a higher dimensional analogue of extriangulated categories defined by Nakaoka-Palu, but also gives a simultaneous generalization of n-exact categories in the sense of Jasso and (n+2)-angulated in the sense of Geiss-Keller-Oppermann. Let C be an n-exangulated category with enough projectives and enough injectives, and X a cluster tilting subcategory of C. In this article, we show that the quotient category C/X is an n-abelian category, and it is equivalent to an n-cluster tilting subcategory of an abelian category with enough projectives. These results generalize the work of Jacobsen-Jørgensen and Zhou-Zhu for (n+2)-angulated categories. Moreover, it highlights new phenomena when it is applied to n-exact categories.

Ladders of recollements of abelian categories

Ladders of recollements of abelian categories are introduced, and used to address three general problems. Ladders of a certain height allow to construct recollements of triangulated categories, involving derived categories and singularity categories, from abelian ones. Ladders also allow to tilt abelian recollements, and ladders guarantee that properties like Gorenstein projective or injective are preserved by some functors in abelian recollements. Breaking symmetry is crucial in developing this theory.

Derived categories of functors and quiver sheaves

Let [Formula: see text] be small category and [Formula: see text] an arbitrary category. Consider the category [Formula: see text] whose objects are functors from [Formula: see text] to [Formula: see text] and whose morphisms are natural transformations. Let [Formula: see text] be another category, and again, consider the category [Formula: see text]. Now, given a functor [Formula: see text] we construct the induced functor [Formula: see text]. Assuming [Formula: see text] and [Formula: see text] to be abelian categories, it follows that the categories [Formula: see text] and [Formula: see text] are also abelian. We have two main goals: first, to find a relationship between the derived category [Formula: see text] and the category [Formula: see text]; second relate the functors [Formula: see text] and [Formula: see text]. We apply the general results obtained to the special case of quiver sheaves.

THE YONEDA EXT AND ARBITRARY COPRODUCTS IN ABELIAN CATEGORIES

AbstractThere are well-known identities involving the Ext bifunctor, coproducts, and products in AB4 abelian categories with enough projectives. Namely, for every such category \[\mathcal{A}\], given an object X and a set of objects \[{\{ {{\text{A}}_{\text{i}}}\} _{{\text{i}} \in {\text{I}}}}\], an isomorphism \[Ext_\mathcal{A}^{\text{n}}({ \oplus _{{\text{i}} \in {\text{I}}}}{{\text{A}}_{\text{i}}},{\text{X}}) \cong \prod\nolimits_{{\text{i}} \in {\text{I}}} {Ext_\mathcal{A}^{\text{n}}({{\text{A}}_{\text{i}}},{\text{X}})} \] can be built, where \[Ex{t^{\text{n}}}\] is the nth derived functor of the Hom functor. The goal of this paper is to show a similar isomorphism for the nth Yoneda Ext, which is a functor equivalent to \[Ex{t^{\text{n}}}\] that can be defined in more general contexts. The desired isomorphism is constructed explicitly by using colimits in AB4 abelian categories with not necessarily enough projectives nor injectives, extending a result by Colpi and Fuller in [8]. Furthermore, the isomorphisms constructed are used to characterize AB4 categories. A dual result is also stated.

Semi-derived Ringel-Hall algebras and Drinfeld double

Let A be an arbitrary hereditary abelian category that may not have enough projective objects. For example, A can be the category of finite-dimensional representations of a quiver or the category of coherent sheaves on a smooth projective curve or on a weighted projective line. Inspired by the works of Bridgeland and Gorsky, we define the semi-derived Ringel-Hall algebra of A, denoted by SDHZ/2(A), to be the localization of a quotient algebra of the Ringel-Hall algebra of the category of Z/2-graded complexes over A.We obtain the following three main results. The semi-derived Ringel-Hall algebra has a natural basis. A twisted version of the semi-derived Ringel-Hall algebra of A is isomorphic to the Drinfeld double of the twisted extended Ringel-Hall algebra Htwe(A) of A. If A has a tilting object T, then its semi-derived Ringel-Hall algebra is isomorphic to the Z/2-graded semi-derived Hall algebra SDHZ/2(addT) of the exact category add T defined by Gorsky, and so is isomorphic to Bridgeland's Hall algebra of mod(End(T)op).

Spanier–Whitehead Categories of Resolving Subcategories and Comparison with Singularity Categories

Let $\mathcal {A}$ be an abelian category with enough projective objects, and let $\mathcal {X}$ be a quasi-resolving subcategory of $\mathcal {A}$ . In this paper, we investigate the affinity of the Spanier–Whitehead category $\mathsf {SW}(\mathcal {X})$ of the stable category of $\mathcal {X}$ with the singularity category $\mathsf {D}_{\mathsf {sg}}(\mathcal {A})$ of $\mathcal {A}$ . We construct a fully faithful triangle functor from $\mathsf {SW}(\mathcal {X})$ to $\mathsf {D}_{\mathsf {sg}}(\mathcal {A})$ , and we prove that it is dense if and only if the bounded derived category $\mathsf {D}^{\mathsf {b}}(\mathcal {A})$ of $\mathcal {A}$ is generated by $\mathcal {X}$ . Applying this result to commutative rings, we obtain characterizations of the isolated singularities, the Gorenstein rings and the Cohen–Macaulay rings. Moreover, we classify the Spanier–Whitehead categories over complete intersections. Finally, we explore a method to compute the (Rouquier) dimension of the triangulated category $\mathsf {SW}(\mathcal {X})$ in terms of generation in $\mathcal {X}$ .

Subprojectivity in Abelian Categories

In the last few years, López-Permouth and several collaborators have introduced a new approach in the study of the classical projectivity, injectivity and flatness of modules. This way, they introduced subprojectivity domains of modules as a tool to measure, somehow, the projectivity level of such a module (so not just to determine whether or not the module is projective). In this paper we develop a new treatment of the subprojectivity in any abelian category which shed more light on some of its various important aspects. Namely, in terms of subprojectivity, some classical results are unified and some classical rings are characterized. It is also shown that, in some categories, the subprojectivity measures notions other than the projectivity. Furthermore, this new approach allows, in addition to establishing nice generalizations of known results, to construct various new examples such as the subprojectivity domain of the class of Gorenstein projective objects, the class of semi-projective complexes and particular types of representations of a finite linear quiver. The paper ends with a study showing that the fact that a subprojectivity domain of a class coincides with its first right Ext-orthogonal class can be characterized in terms of the existence of preenvelopes and precovers.

Intersections, sums, and the Jordan-Hölder property for exact categories

We investigate how the concepts of intersection and sums of subobjects carry to exact categories. We obtain a new characterisation of quasi-abelian categories in terms of admitting admissible intersections in the sense of [23] . There are also many alternative characterisations of abelian categories as those that additionally admit admissible sums and in terms of properties of admissible morphisms . We then define a generalised notion of intersection and sum which every exact category admits. Using these new notions, we define and study classes of exact categories that satisfy the Jordan-Hölder property for exact categories, namely the Diamond exact categories and Artin-Wedderburn exact categories. By explicitly describing all exact structures on A = rep Λ for a Nakayama algebra Λ we characterise all Artin-Wedderburn exact structures on A and show that these are precisely the exact structures with the Jordan-Hölder property.

Irregular perverse sheaves

We introduce irregular constructible sheaves, which are ${\mathbb {C}}$-constructible with coefficients in a finite version of the Novikov ring $\Lambda$ and special gradings. We show that the bounded derived category of cohomologically irregular constructible complexes is equivalent to the bounded derived category of holonomic ${\mathcal {D}}$-modules by a modification of D’Agnolo and Kashiwara's irregular Riemann–Hilbert correspondence. The bounded derived category of cohomologically irregular constructible complexes is equipped with the irregular perverse $t$-structure, which is a straightforward generalization of usual perverse $t$-structure, and we prove that its heart is equivalent to the abelian category of holonomic ${\mathcal {D}}$-modules. We also develop the algebraic version of the theory.

Model Structures on the Category of Complexes of Quiver Representations

In this paper, we study the category \(\mathbb {C} (\mathrm{{Rep}}(\mathcal {Q}, \mathfrak {A}))\) of complexes of representations of quiver \(\mathcal {Q}\) with values in an abelian category \(\mathfrak {A}\). We develop a method for constructing some model structures on \(\mathbb {C} (\mathrm{{Rep}}(\mathcal {Q}, \mathfrak {A}))\) based on componentwise notion. Moreover, we also show that these model structures are monoidal. As an application of these model structures, we introduce some descriptions of the derived category of complexes of representations of \(\mathcal {Q}\) in \(\mathrm{Mod\hbox {-}}R\).

A counterexample to vanishing conjectures for negative K-theory

In a 2006 article Schlichting conjectured that the negative {\it K--}theory of any abelian category must vanish. This conjecture was generalized in a 2019 article by Antieau, Gepner and Heller, who hypothesized that the negative {\it K--}theory of any category with a bounded {\it t--}structure must vanish. Both conjectures will be shown to be false.

Integrals Along Bimonoid Homomorphisms

We introduce a notion of an integral along a bimonoid homomorphism as a simultaneous generalization of the integral and cointegral of bimonoids. The purpose of this paper is to characterize an existence of a specific integral, called a normalized generator integral, along a bimonoid homomorphism in terms of the kernel and cokernel of the homomorphism. We introduce a notion of a volume on an abelian category as a generalization of the dimension of vector spaces and the order of abelian groups. In applications, we show that there exists a nontrivial volume partially defined on a category of bicommutative Hopf monoids. The volume yields a notion of Fredholm homomorphisms between bicommutative Hopf monoids, which gives an analogue of the Fredholm index theory. This paper gives a technical preliminary of our subsequent paper about a construction of TQFT’s.

Homological Dimensions Relative to Special Subcategories

Let [Formula: see text] be an abelian category, [Formula: see text] an additive, full and self-orthogonal subcategory of [Formula: see text] closed under direct summands, [Formula: see text] the right Gorenstein subcategory of [Formula: see text] relative to [Formula: see text], and [Formula: see text] the left orthogonal class of [Formula: see text]. For an object [Formula: see text] in [Formula: see text], we prove that if [Formula: see text] is in the right 1-orthogonal class of [Formula: see text], then the [Formula: see text]-projective and [Formula: see text]-projective dimensions of [Formula: see text] are identical; if the [Formula: see text]-projective dimension of [Formula: see text] is finite, then the [Formula: see text]-projective and [Formula: see text]-projective dimensions of [Formula: see text] are identical. We also prove that the supremum of the [Formula: see text]-projective dimensions of objects with finite [Formula: see text]-projective dimension and that of the [Formula: see text]-projective dimensions of objects with finite [Formula: see text]-projective dimension coincide. Then we apply these results to the category of modules.

Piecewise Hereditary Triangular Matrix Algebras

For any positive integer [Formula: see text], we clearly describe all finite-dimensional algebras [Formula: see text] such that the upper triangular matrix algebras [Formula: see text] are piecewise hereditary. Consequently, we describe all finite-dimensional algebras [Formula: see text] such that their derived categories of [Formula: see text]-complexes are triangulated equivalent to derived categories of hereditary abelian categories, and we describe the tensor algebras [Formula: see text] for which their singularity categories are triangulated orbit categories of the derived categories of hereditary abelian categories.

Covers and direct limits: a contramodule-based approach

We present applications of contramodule techniques to the Enochs conjecture about covers and direct limits, both in the categorical tilting context and beyond. In the $n$-tilting-cotilting correspondence situation, if $\mathsf A$ is a Grothendieck abelian category and the related abelian category $\mathsf B$ is equivalent to the category of contramodules over a topological ring $\mathfrak R$ belonging to one of certain four classes of topological rings (e.g., $\mathfrak R$ is commutative), then the left tilting class is covering in $\mathsf A$ if and only if it is closed under direct limits in $\mathsf A$, and if and only if all the discrete quotient rings of the topological ring $\mathfrak R$ are perfect. More generally, if $M$ is a module satisfying a certain telescope Hom exactness condition (e.g., $M$ is $\Sigma$-pure-$\operatorname{Ext}^1$-self-orthogonal) and the topological ring $\mathfrak R$ of endomorphisms of $M$ belongs to one of certain seven classes of topological rings, then the class $\mathsf{Add}(M)$ is closed under direct limits if and only if every countable direct limit of copies of $M$ has an $\mathsf{Add}(M)$-cover, and if and only if $M$ has perfect decomposition. In full generality, for an additive category $\mathsf A$ with (co)kernels and a precovering class $\mathsf L\subset\mathsf A$ closed under summands, an object $N\in\mathsf A$ has an $\mathsf L$-cover if and only if a certain object $\Psi(N)$ in an abelian category $\mathsf B$ with enough projectives has a projective cover. The $1$-tilting modules and objects arising from injective ring epimorphisms of projective dimension $1$ form a class of examples which we discuss.

Extension groups between atoms in abelian categories

We introduce the extension groups between atoms in an abelian category. For a locally noetherian Grothendieck category, the localizing subcategories closed under injective envelopes are characterized in terms of those extension groups. We also introduce the virtual duals of the extension groups between atoms to measure the global dimension of the category. A new topological property of atom spectra is revealed and it is used to relate the projective dimensions of atoms with the Krull-Gabriel dimensions. As a byproduct of the topological observation, we show that there exists a spectral space that is not homeomorphic to the atom spectrum of any abelian category.

On Noetherian schemes over (\mathcal{C},\otimes,1)$ and the category of quasi-coherent sheaves

Let $(\mathcal C,\otimes,1)$ be an abelian symmetric monoidal category satisfying certain conditions and let $X$ be a scheme over $(\mathcal C,\otimes,1)$ in the sense of Toen and Vaquie. In this paper we show that when $X$ is quasi-compact and semi-separated, any quasi-coherent sheaf on $X$ may be expressed as a directed colimit of its finitely generated quasi-coherent submodules. Thereafter, we introduce a notion of objects in $(\mathcal C,\otimes,1)$ that satisfy several properties similar to those of fields in usual commutative algebra. Finally we show that the points of a Noetherian, quasi-compact and semi-separated scheme $X$ over such a field object $K$ in $(\mathcal C,\otimes,1)$ can be recovered from certain kinds of functors between categories of quasi-coherent sheaves. The latter is a partial generalization of some recent results of Brandenburg and Chirvasitu.

The special fiber of the motivic deformation of the stable homotopy category is algebraic

For each prime $p$, we define a $t$-structure on the category $\widehat{S^{0,0}}/\tau\text{-}\mathbf{Mod}_{harm}^b$ of harmonic $\mathbb{C}$-motivic left module spectra over $\widehat{S^{0,0}}/\tau$, whose MGL-homology has bounded Chow-Novikov degree, such that its heart is equivalent to the abelian category of $p$-completed $BP_*BP$-comodules that are concentrated in even degrees. We prove that $\widehat{S^{0,0}}/\tau\text{-}\mathbf{Mod}_{harm}^b$ is equivalent to $\mathcal{D}^b({{BP}_*{BP}\text{-}\mathbf{Comod}}^{ev})$ as stable $\infty$-categories equipped with $t$-structures. As an application, for each prime $p$, we prove that the motivic Adams spectral sequence for $\widehat{S^{0,0}}/\tau$, which converges to the motivic homotopy groups of $\widehat{S^{0,0}}/\tau$, is isomorphic to the algebraic Novikov spectral sequence, which converges to the classical Adams-Novikov $E_2$-page for the sphere spectrum $\widehat{S^0}$. This isomorphism of spectral sequences allows Isaksen and the second and third authors to compute the stable homotopy groups of spheres at least to the 90-stem, with ongoing computations into even higher dimensions.

Finitistic Homological Dimensions Relative to Subcategories

Let C⊆T be subcategories of an abelian category A. Under some certain conditions, we show that the C-finitistic and T-finitistic global dimensions of A are identical. Some applications are given; in particular, some known results are obtained as corollaries.

Cartier modules and cyclotomic spectra

We construct and study a t t -structure on p p -typical cyclotomic spectra and explain how to recover crystalline cohomology of smooth schemes over perfect fields using this t t -structure. Our main tool is a new approach to p p -typical cyclotomic spectra via objects we call p p -typical topological Cartier modules. Using these, we prove that the heart of the cyclotomic t t -structure is the full subcategory of derived V V -complete objects in the abelian category of p p -typical Cartier modules.